From 017644609992aa2aa9a6a6db0d9f692c4739110e Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Wed, 2 Feb 2022 16:23:27 +0000 Subject: Update on Overleaf. --- iopams.sty | 88 ++++++++++++++++++++++++++++++++++++- iopart10.clo | 125 +++++++++++++++++++++++++++++++++++++++++++++++++++- iopart12.clo | 126 ++++++++++++++++++++++++++++++++++++++++++++++++++++- setstack.sty | 140 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++- stokes.tex | 44 +++++++++++++++++++ 5 files changed, 519 insertions(+), 4 deletions(-) diff --git a/iopams.sty b/iopams.sty index baf2b72..044dde9 100644 --- a/iopams.sty +++ b/iopams.sty @@ -1 +1,87 @@ -%% %% This is file `iopams.sty' %% File to include AMS fonts and extra definitions for bold greek %% characters for use with iopart.cls %% \NeedsTeXFormat{LaTeX2e} \ProvidesPackage{iopams}[1997/02/13 v1.0] \RequirePackage{amsgen}[1995/01/01] \RequirePackage{amsfonts}[1995/01/01] \RequirePackage{amssymb}[1995/01/01] \RequirePackage{amsbsy}[1995/01/01] % \iopamstrue % \newif\ifiopams in iopart.cls & iopbk2e.cls % % allows optional text to be in author guidelines % % Bold lower case Greek letters % \newcommand{\balpha}{\boldsymbol{\alpha}} \newcommand{\bbeta}{\boldsymbol{\beta}} \newcommand{\bgamma}{\boldsymbol{\gamma}} \newcommand{\bdelta}{\boldsymbol{\delta}} \newcommand{\bepsilon}{\boldsymbol{\epsilon}} \newcommand{\bzeta}{\boldsymbol{\zeta}} \newcommand{\bfeta}{\boldsymbol{\eta}} \newcommand{\btheta}{\boldsymbol{\theta}} \newcommand{\biota}{\boldsymbol{\iota}} \newcommand{\bkappa}{\boldsymbol{\kappa}} \newcommand{\blambda}{\boldsymbol{\lambda}} \newcommand{\bmu}{\boldsymbol{\mu}} \newcommand{\bnu}{\boldsymbol{\nu}} \newcommand{\bxi}{\boldsymbol{\xi}} \newcommand{\bpi}{\boldsymbol{\pi}} \newcommand{\brho}{\boldsymbol{\rho}} \newcommand{\bsigma}{\boldsymbol{\sigma}} \newcommand{\btau}{\boldsymbol{\tau}} \newcommand{\bupsilon}{\boldsymbol{\upsilon}} \newcommand{\bphi}{\boldsymbol{\phi}} \newcommand{\bchi}{\boldsymbol{\chi}} \newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bomega}{\boldsymbol{\omega}} \newcommand{\bvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\bvartheta}{\boldsymbol{\vartheta}} \newcommand{\bvaromega}{\boldsymbol{\varomega}} \newcommand{\bvarrho}{\boldsymbol{\varrho}} \newcommand{\bvarzeta}{\boldsymbol{\varsigma}} %NB really sigma \newcommand{\bvarsigma}{\boldsymbol{\varsigma}} \newcommand{\bvarphi}{\boldsymbol{\varphi}} % % Bold upright capital Greek letters % \newcommand{\bGamma}{\boldsymbol{\Gamma}} \newcommand{\bDelta}{\boldsymbol{\Delta}} \newcommand{\bTheta}{\boldsymbol{\Theta}} \newcommand{\bLambda}{\boldsymbol{\Lambda}} \newcommand{\bXi}{\boldsymbol{\Xi}} \newcommand{\bPi}{\boldsymbol{\Pi}} \newcommand{\bSigma}{\boldsymbol{\Sigma}} \newcommand{\bUpsilon}{\boldsymbol{\Upsilon}} \newcommand{\bPhi}{\boldsymbol{\Phi}} \newcommand{\bPsi}{\boldsymbol{\Psi}} \newcommand{\bOmega}{\boldsymbol{\Omega}} % % Bold versions of miscellaneous symbols % \newcommand{\bpartial}{\boldsymbol{\partial}} \newcommand{\bell}{\boldsymbol{\ell}} \newcommand{\bimath}{\boldsymbol{\imath}} \newcommand{\bjmath}{\boldsymbol{\jmath}} \newcommand{\binfty}{\boldsymbol{\infty}} \newcommand{\bnabla}{\boldsymbol{\nabla}} \newcommand{\bdot}{\boldsymbol{\cdot}} % % Symbols for caption % \renewcommand{\opensquare}{\mbox{$\square$}} \renewcommand{\opentriangle}{\mbox{$\vartriangle$}} \renewcommand{\opentriangledown}{\mbox{$\triangledown$}} \renewcommand{\opendiamond}{\mbox{$\lozenge$}} \renewcommand{\fullsquare}{\mbox{$\blacksquare$}} \newcommand{\fulldiamond}{\mbox{$\blacklozenge$}} \newcommand{\fullstar}{\mbox{$\bigstar$}} \newcommand{\fulltriangle}{\mbox{$\blacktriangle$}} \newcommand{\fulltriangledown}{\mbox{$\blacktriangledown$}} \endinput %% %% End of file `iopams.sty'. \ No newline at end of file +%% +%% This is file `iopams.sty' +%% File to include AMS fonts and extra definitions for bold greek +%% characters for use with iopart.cls +%% +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{iopams}[1997/02/13 v1.0] +\RequirePackage{amsgen}[1995/01/01] +\RequirePackage{amsfonts}[1995/01/01] +\RequirePackage{amssymb}[1995/01/01] +\RequirePackage{amsbsy}[1995/01/01] +% +\iopamstrue % \newif\ifiopams in iopart.cls & iopbk2e.cls +% % allows optional text to be in author guidelines +% +% Bold lower case Greek letters +% +\newcommand{\balpha}{\boldsymbol{\alpha}} +\newcommand{\bbeta}{\boldsymbol{\beta}} +\newcommand{\bgamma}{\boldsymbol{\gamma}} +\newcommand{\bdelta}{\boldsymbol{\delta}} +\newcommand{\bepsilon}{\boldsymbol{\epsilon}} +\newcommand{\bzeta}{\boldsymbol{\zeta}} +\newcommand{\bfeta}{\boldsymbol{\eta}} +\newcommand{\btheta}{\boldsymbol{\theta}} +\newcommand{\biota}{\boldsymbol{\iota}} +\newcommand{\bkappa}{\boldsymbol{\kappa}} +\newcommand{\blambda}{\boldsymbol{\lambda}} +\newcommand{\bmu}{\boldsymbol{\mu}} +\newcommand{\bnu}{\boldsymbol{\nu}} +\newcommand{\bxi}{\boldsymbol{\xi}} +\newcommand{\bpi}{\boldsymbol{\pi}} +\newcommand{\brho}{\boldsymbol{\rho}} +\newcommand{\bsigma}{\boldsymbol{\sigma}} +\newcommand{\btau}{\boldsymbol{\tau}} +\newcommand{\bupsilon}{\boldsymbol{\upsilon}} +\newcommand{\bphi}{\boldsymbol{\phi}} +\newcommand{\bchi}{\boldsymbol{\chi}} +\newcommand{\bpsi}{\boldsymbol{\psi}} +\newcommand{\bomega}{\boldsymbol{\omega}} +\newcommand{\bvarepsilon}{\boldsymbol{\varepsilon}} +\newcommand{\bvartheta}{\boldsymbol{\vartheta}} +\newcommand{\bvaromega}{\boldsymbol{\varomega}} +\newcommand{\bvarrho}{\boldsymbol{\varrho}} +\newcommand{\bvarzeta}{\boldsymbol{\varsigma}} %NB really sigma +\newcommand{\bvarsigma}{\boldsymbol{\varsigma}} +\newcommand{\bvarphi}{\boldsymbol{\varphi}} +% +% Bold upright capital Greek letters +% +\newcommand{\bGamma}{\boldsymbol{\Gamma}} +\newcommand{\bDelta}{\boldsymbol{\Delta}} +\newcommand{\bTheta}{\boldsymbol{\Theta}} +\newcommand{\bLambda}{\boldsymbol{\Lambda}} +\newcommand{\bXi}{\boldsymbol{\Xi}} +\newcommand{\bPi}{\boldsymbol{\Pi}} +\newcommand{\bSigma}{\boldsymbol{\Sigma}} +\newcommand{\bUpsilon}{\boldsymbol{\Upsilon}} +\newcommand{\bPhi}{\boldsymbol{\Phi}} +\newcommand{\bPsi}{\boldsymbol{\Psi}} +\newcommand{\bOmega}{\boldsymbol{\Omega}} +% +% Bold versions of miscellaneous symbols +% +\newcommand{\bpartial}{\boldsymbol{\partial}} +\newcommand{\bell}{\boldsymbol{\ell}} +\newcommand{\bimath}{\boldsymbol{\imath}} +\newcommand{\bjmath}{\boldsymbol{\jmath}} +\newcommand{\binfty}{\boldsymbol{\infty}} +\newcommand{\bnabla}{\boldsymbol{\nabla}} +\newcommand{\bdot}{\boldsymbol{\cdot}} +% +% Symbols for caption +% +\renewcommand{\opensquare}{\mbox{$\square$}} +\renewcommand{\opentriangle}{\mbox{$\vartriangle$}} +\renewcommand{\opentriangledown}{\mbox{$\triangledown$}} +\renewcommand{\opendiamond}{\mbox{$\lozenge$}} +\renewcommand{\fullsquare}{\mbox{$\blacksquare$}} +\newcommand{\fulldiamond}{\mbox{$\blacklozenge$}} +\newcommand{\fullstar}{\mbox{$\bigstar$}} +\newcommand{\fulltriangle}{\mbox{$\blacktriangle$}} +\newcommand{\fulltriangledown}{\mbox{$\blacktriangledown$}} + +\endinput +%% +%% End of file `iopams.sty'. diff --git a/iopart10.clo b/iopart10.clo index 1805a0b..dec5747 100644 --- a/iopart10.clo +++ b/iopart10.clo @@ -1 +1,124 @@ -%% %% This is file `iopart10.clo' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \ProvidesFile{iopart10.clo}[1997/01/13 v1.0 IOP Book file (size option)] \renewcommand\normalsize{% \@setfontsize\normalsize\@xpt\@xiipt \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI} \normalsize \newcommand\small{% \@setfontsize\small\@ixpt{11}% \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ \abovedisplayshortskip \z@ \@plus2\p@ \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 4\p@ \@plus2\p@ \@minus2\p@ \parsep 2\p@ \@plus\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\footnotesize{% \@setfontsize\footnotesize\@viiipt{9.5}% \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@ \abovedisplayshortskip \z@ \@plus\p@ \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 3\p@ \@plus\p@ \@minus\p@ \parsep 2\p@ \@plus\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt} \newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt} \newcommand\large{\@setfontsize\large\@xiipt{14}} \newcommand\Large{\@setfontsize\Large\@xivpt{18}} \newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}} \newcommand\huge{\@setfontsize\huge\@xxpt{25}} \newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}} \if@twocolumn \setlength\parindent{12\p@} \else \setlength\parindent{15\p@} \fi \setlength\headheight{12\p@} \setlength\headsep {12\p@} \setlength\topskip {10\p@} \setlength\footskip{20\p@} \setlength\maxdepth{.5\topskip} \setlength\@maxdepth\maxdepth \setlength\textwidth{31pc} \setlength\textheight{49pc} \setlength\oddsidemargin {24\p@} \setlength\evensidemargin {24\p@} \setlength\marginparwidth {72\p@} \setlength\marginparsep {10\p@} \setlength\marginparpush{5\p@} \setlength\topmargin{\z@} \setlength\footnotesep{6.65\p@} \setlength{\skip\footins} {9\p@ \@plus 4\p@ \@minus 2\p@} \setlength\floatsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\textfloatsep {20\p@ \@plus 2\p@ \@minus 4\p@} \setlength\intextsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\dblfloatsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@} \setlength\@fptop{0\p@} \setlength\@fpsep{8\p@ \@plus 2fil} \setlength\@fpbot{0\p@} \setlength\@dblfptop{0\p@} \setlength\@dblfpsep{8\p@ \@plus 2fil} \setlength\@dblfpbot{0\p@} \setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@} \def\@listI{\leftmargin\leftmargini \parsep=\z@ \topsep=5\p@ \@plus3\p@ \@minus3\p@ \itemsep=3\p@ \@plus2\p@ \@minus\p@} \let\@listi\@listI \@listi \def\@listii {\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep=2\p@ \@plus2\p@ \@minus\p@ \parsep=\z@ \itemsep=\parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep \topsep=\z@ \parsep=\z@ \partopsep=\z@ \itemsep=\z@} \def\@listiv {\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv {\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi {\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \endinput %% %% End of file `iopart.clo'. \ No newline at end of file +%% +%% This is file `iopart10.clo' +%% +%% This file is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. +%% +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! Double quote \" Hash (number) \# +%% Dollar \$ Percent \% Ampersand \& +%% Acute accent \' Left paren \( Right paren \) +%% Asterisk \* Plus \+ Comma \, +%% Minus \- Point \. Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} +\ProvidesFile{iopart10.clo}[1997/01/13 v1.0 IOP Book file (size option)] +\renewcommand\normalsize{% + \@setfontsize\normalsize\@xpt\@xiipt + \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ + \belowdisplayskip \abovedisplayskip + \let\@listi\@listI} +\normalsize +\newcommand\small{% + \@setfontsize\small\@ixpt{11}% + \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus2\p@ + \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 4\p@ \@plus2\p@ \@minus2\p@ + \parsep 2\p@ \@plus\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\footnotesize{% + \@setfontsize\footnotesize\@viiipt{9.5}% + \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus\p@ + \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 3\p@ \@plus\p@ \@minus\p@ + \parsep 2\p@ \@plus\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt} +\newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt} +\newcommand\large{\@setfontsize\large\@xiipt{14}} +\newcommand\Large{\@setfontsize\Large\@xivpt{18}} +\newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}} +\newcommand\huge{\@setfontsize\huge\@xxpt{25}} +\newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}} +\if@twocolumn + \setlength\parindent{12\p@} +\else + \setlength\parindent{15\p@} +\fi +\setlength\headheight{12\p@} +\setlength\headsep {12\p@} +\setlength\topskip {10\p@} +\setlength\footskip{20\p@} +\setlength\maxdepth{.5\topskip} +\setlength\@maxdepth\maxdepth +\setlength\textwidth{31pc} +\setlength\textheight{49pc} +\setlength\oddsidemargin {24\p@} +\setlength\evensidemargin {24\p@} +\setlength\marginparwidth {72\p@} +\setlength\marginparsep {10\p@} +\setlength\marginparpush{5\p@} +\setlength\topmargin{\z@} +\setlength\footnotesep{6.65\p@} +\setlength{\skip\footins} {9\p@ \@plus 4\p@ \@minus 2\p@} +\setlength\floatsep {12\p@ \@plus 2\p@ \@minus 2\p@} +\setlength\textfloatsep {20\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\intextsep {12\p@ \@plus 2\p@ \@minus 2\p@} +\setlength\dblfloatsep {12\p@ \@plus 2\p@ \@minus 2\p@} +\setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\@fptop{0\p@} +\setlength\@fpsep{8\p@ \@plus 2fil} +\setlength\@fpbot{0\p@} +\setlength\@dblfptop{0\p@} +\setlength\@dblfpsep{8\p@ \@plus 2fil} +\setlength\@dblfpbot{0\p@} +\setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@} +\def\@listI{\leftmargin\leftmargini + \parsep=\z@ + \topsep=5\p@ \@plus3\p@ \@minus3\p@ + \itemsep=3\p@ \@plus2\p@ \@minus\p@} +\let\@listi\@listI +\@listi +\def\@listii {\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth-\labelsep + \topsep=2\p@ \@plus2\p@ \@minus\p@ + \parsep=\z@ + \itemsep=\parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii + \advance\labelwidth-\labelsep + \topsep=\z@ + \parsep=\z@ + \partopsep=\z@ + \itemsep=\z@} +\def\@listiv {\leftmargin\leftmarginiv + \labelwidth\leftmarginiv + \advance\labelwidth-\labelsep} +\def\@listv {\leftmargin\leftmarginv + \labelwidth\leftmarginv + \advance\labelwidth-\labelsep} +\def\@listvi {\leftmargin\leftmarginvi + \labelwidth\leftmarginvi + \advance\labelwidth-\labelsep} +\endinput +%% +%% End of file `iopart.clo'. diff --git a/iopart12.clo b/iopart12.clo index 48ec5c1..d392bf9 100644 --- a/iopart12.clo +++ b/iopart12.clo @@ -1 +1,125 @@ -%% %% This is file `iopart12.clo' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \ProvidesFile{iopart12.clo}[1997/01/15 v1.0 LaTeX2e file (size option)] \renewcommand\normalsize{% \@setfontsize\normalsize\@xiipt{16}% \abovedisplayskip 12\p@ \@plus3\p@ \@minus7\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI} \normalsize \newcommand\small{% \@setfontsize\small\@xipt{14}% \abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \def\@listi{\leftmargin\leftmargini \topsep 9\p@ \@plus3\p@ \@minus5\p@ \parsep 4.5\p@ \@plus2\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\footnotesize{% % \@setfontsize\footnotesize\@xpt\@xiipt \@setfontsize\footnotesize\@xpt{13}% \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ \def\@listi{\leftmargin\leftmargini \topsep 6\p@ \@plus2\p@ \@minus2\p@ \parsep 3\p@ \@plus2\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9.5}} \newcommand\tiny{\@setfontsize\tiny\@vipt\@viipt} \newcommand\large{\@setfontsize\large\@xivpt{18}} \newcommand\Large{\@setfontsize\Large\@xviipt{22}} \newcommand\LARGE{\@setfontsize\LARGE\@xxpt{25}} \newcommand\huge{\@setfontsize\huge\@xxvpt{30}} \let\Huge=\huge \if@twocolumn \setlength\parindent{14\p@} \else \setlength\parindent{18\p@} \fi \setlength\headheight{14\p@} \setlength\headsep{14\p@} \setlength\topskip{12\p@} \setlength\footskip{24\p@} \setlength\maxdepth{.5\topskip} \setlength\@maxdepth\maxdepth \setlength\textwidth{37.2pc} \setlength\textheight{56pc} \setlength\oddsidemargin {\z@} \setlength\evensidemargin {\z@} \setlength\marginparwidth {72\p@} \setlength\marginparsep{10\p@} \setlength\marginparpush{5\p@} \setlength\topmargin{-12pt} \setlength\footnotesep{8.4\p@} \setlength{\skip\footins} {10.8\p@ \@plus 4\p@ \@minus 2\p@} \setlength\floatsep {14\p@ \@plus 2\p@ \@minus 4\p@} \setlength\textfloatsep {24\p@ \@plus 2\p@ \@minus 4\p@} \setlength\intextsep {16\p@ \@plus 4\p@ \@minus 4\p@} \setlength\dblfloatsep {16\p@ \@plus 2\p@ \@minus 4\p@} \setlength\dbltextfloatsep{24\p@ \@plus 2\p@ \@minus 4\p@} \setlength\@fptop{0\p@} \setlength\@fpsep{10\p@ \@plus 1fil} \setlength\@fpbot{0\p@} \setlength\@dblfptop{0\p@} \setlength\@dblfpsep{10\p@ \@plus 1fil} \setlength\@dblfpbot{0\p@} \setlength\partopsep{3\p@ \@plus 2\p@ \@minus 2\p@} \def\@listI{\leftmargin\leftmargini \parsep=\z@ \topsep=6\p@ \@plus3\p@ \@minus3\p@ \itemsep=3\p@ \@plus2\p@ \@minus1\p@} \let\@listi\@listI \@listi \def\@listii {\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep=3\p@ \@plus2\p@ \@minus\p@ \parsep=\z@ \itemsep=\parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep \topsep=\z@ \parsep=\z@ \partopsep=\z@ \itemsep=\z@} \def\@listiv {\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv{\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi {\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \endinput %% %% End of file `iopart12.clo'. \ No newline at end of file +%% +%% This is file `iopart12.clo' +%% +%% This file is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. +%% +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! Double quote \" Hash (number) \# +%% Dollar \$ Percent \% Ampersand \& +%% Acute accent \' Left paren \( Right paren \) +%% Asterisk \* Plus \+ Comma \, +%% Minus \- Point \. Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} +\ProvidesFile{iopart12.clo}[1997/01/15 v1.0 LaTeX2e file (size option)] +\renewcommand\normalsize{% + \@setfontsize\normalsize\@xiipt{16}% + \abovedisplayskip 12\p@ \@plus3\p@ \@minus7\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ + \belowdisplayskip \abovedisplayskip + \let\@listi\@listI} +\normalsize +\newcommand\small{% + \@setfontsize\small\@xipt{14}% + \abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 9\p@ \@plus3\p@ \@minus5\p@ + \parsep 4.5\p@ \@plus2\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\footnotesize{% +% \@setfontsize\footnotesize\@xpt\@xiipt + \@setfontsize\footnotesize\@xpt{13}% + \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 6\p@ \@plus2\p@ \@minus2\p@ + \parsep 3\p@ \@plus2\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9.5}} +\newcommand\tiny{\@setfontsize\tiny\@vipt\@viipt} +\newcommand\large{\@setfontsize\large\@xivpt{18}} +\newcommand\Large{\@setfontsize\Large\@xviipt{22}} +\newcommand\LARGE{\@setfontsize\LARGE\@xxpt{25}} +\newcommand\huge{\@setfontsize\huge\@xxvpt{30}} +\let\Huge=\huge +\if@twocolumn + \setlength\parindent{14\p@} +\else + \setlength\parindent{18\p@} +\fi +\setlength\headheight{14\p@} +\setlength\headsep{14\p@} +\setlength\topskip{12\p@} +\setlength\footskip{24\p@} +\setlength\maxdepth{.5\topskip} +\setlength\@maxdepth\maxdepth +\setlength\textwidth{37.2pc} +\setlength\textheight{56pc} +\setlength\oddsidemargin {\z@} +\setlength\evensidemargin {\z@} +\setlength\marginparwidth {72\p@} +\setlength\marginparsep{10\p@} +\setlength\marginparpush{5\p@} +\setlength\topmargin{-12pt} +\setlength\footnotesep{8.4\p@} +\setlength{\skip\footins} {10.8\p@ \@plus 4\p@ \@minus 2\p@} +\setlength\floatsep {14\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\textfloatsep {24\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\intextsep {16\p@ \@plus 4\p@ \@minus 4\p@} +\setlength\dblfloatsep {16\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\dbltextfloatsep{24\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\@fptop{0\p@} +\setlength\@fpsep{10\p@ \@plus 1fil} +\setlength\@fpbot{0\p@} +\setlength\@dblfptop{0\p@} +\setlength\@dblfpsep{10\p@ \@plus 1fil} +\setlength\@dblfpbot{0\p@} +\setlength\partopsep{3\p@ \@plus 2\p@ \@minus 2\p@} +\def\@listI{\leftmargin\leftmargini + \parsep=\z@ + \topsep=6\p@ \@plus3\p@ \@minus3\p@ + \itemsep=3\p@ \@plus2\p@ \@minus1\p@} +\let\@listi\@listI +\@listi +\def\@listii {\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth-\labelsep + \topsep=3\p@ \@plus2\p@ \@minus\p@ + \parsep=\z@ + \itemsep=\parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii + \advance\labelwidth-\labelsep + \topsep=\z@ + \parsep=\z@ + \partopsep=\z@ + \itemsep=\z@} +\def\@listiv {\leftmargin\leftmarginiv + \labelwidth\leftmarginiv + \advance\labelwidth-\labelsep} +\def\@listv{\leftmargin\leftmarginv + \labelwidth\leftmarginv + \advance\labelwidth-\labelsep} +\def\@listvi {\leftmargin\leftmarginvi + \labelwidth\leftmarginvi + \advance\labelwidth-\labelsep} +\endinput +%% +%% End of file `iopart12.clo'. diff --git a/setstack.sty b/setstack.sty index 1d0130f..da428a6 100644 --- a/setstack.sty +++ b/setstack.sty @@ -1 +1,139 @@ -%% %% This is file `setstack.sty', created by VIK 15 Dec 1998 %% Reproduces useful macros from amsmath.sty, thus avoiding the need to load the entire %% amsmath.sty and run into conflicts %% Adds definitions for \overset, \underset, \sideset, \substack, \boxed, \leftroot, %% \uproot, \dddot, \ddddot, \varrow, \harrow (see The LateX Companion, pp 225-227) \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1998/12/15]% LaTeX date must December 1998 or later \ProvidesPackage{setstack} \DeclareRobustCommand{\text}{% \ifmmode\expandafter\text@\else\expandafter\mbox\fi} \let\nfss@text\text \def\text@#1{\mathchoice {\textdef@\displaystyle\f@size{#1}}% {\textdef@\textstyle\tf@size{\firstchoice@false #1}}% {\textdef@\textstyle\sf@size{\firstchoice@false #1}}% {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}% \check@mathfonts } \def\textdef@#1#2#3{\hbox{{% \everymath{#1}% \let\f@size#2\selectfont #3}}} % adds underset, overset, sideset and % substack features from amsmath.sty (Companion p. 226) \def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag} \def\dft@tag{\invalid@tag{\string\tag\space not allowed here}} \def\default@tag{\let\tag\dft@tag} \def\Let@{\let\\\math@cr} \def\restore@math@cr{\def\math@cr@@@{\cr}} \def\overset#1#2{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}} \def\underset#1#2{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}} \def\sideset#1#2#3{% \@mathmeasure\z@\displaystyle{#3}% \global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@ \setbox\tw@\box\@ne \@mathmeasure4\displaystyle{\copy\tw@#1}% \@mathmeasure6\displaystyle{#3\nolimits#2}% \dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@ \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}% } \newenvironment{subarray}[1]{% \vcenter\bgroup \Let@ \restore@math@cr \default@tag \baselineskip\fontdimen10 \scriptfont\tw@ \advance\baselineskip\fontdimen12 \scriptfont\tw@ \lineskip\thr@@\fontdimen8 \scriptfont\thr@@ \lineskiplimit\lineskip \ialign\bgroup\ifx c#1\hfil\fi $\m@th\scriptstyle##$\hfil\crcr }{% \crcr\egroup\egroup } \newcommand{\substack}[1]{\subarray{c}#1\endsubarray} % definitions of dddot and ddddot(p. 225) \def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont ...}\vss}}}} \def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont....}\vss}}}} % definitions of leftroot, uproot (p.225) \begingroup \catcode`\"=12 \gdef\@@sqrt#1{\radical"270370 {#1}} \endgroup \def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha} \def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha} \newcount\uproot@ \newcount\leftroot@ \def\root{\relaxnext@ \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else \ifx\@let@token\leftroot\let\next@\nextiii@\else \let\next@\plainroot@\fi\fi\next@}% \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}% \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else \DN@.{\FN@\nextv@}\fi\next@.}% \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else \let\next@\plainroot@\fi\next@}% \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}% \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}% \def\nextvii@{\ifx\@let@token\@sptoken \DN@. {\FN@\nextviii@}\else \DN@.{\FN@\nextviii@}\fi\next@.}% \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else \let\next@\plainroot@\fi\next@}% \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}% \bgroup\uproot@\z@\leftroot@\z@\FN@\next@} \def\plainroot@#1\of#2{\setbox\rootbox\hbox{% $\m@th\scriptscriptstyle{#1}$}% \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}} {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup} \def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}% \dimen@\ht\z@\advance\dimen@-\dp\z@ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}% \advance\dimen@ by1.667\wd\@ne \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox \mkern-10mu\mkern\leftroot@ mu\boxz@} % definition of \boxed (math in frame, no dollars) p 225 \def\boxed#1{\fbox{\m@th$\displaystyle#1$}} % definition of \smash for top and bottom parts of expression in braces (p.227) \renewcommand{\smash}[2][tb]{% \def\smash@{#1}% \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else \@xp\makesm@sh\fi{#2}} % additional difinitions for arrows % zero mm wide #2 mm long vertical arrow shifted 1 mm to left, 1 mm up \newcommand{\varrow}[2]{% \unitlength=1mm \begin{picture}(0,6) % (0,0) % - for #1 to point arrow down,+ to point arrow up \end{picture}% \put(0,6){\vector(0,#1 3){#2}} % 6,- for down, 0, + for up } % 1 mm high #2 mm long horizontal arrow shifted 1 mm up (0,-1) \newcommand{\harrow}[2]{% \unitlength=1mm \begin{picture}(8,1)(0,-1) % % 1 mm distance between arrow and stackreled object over it (8,1) \put(0,0){\vector(#1 2,0){#2}} % - to point arrow left, + right \end{picture}% } \endinput %% end of setstack.sty %% Corrections history: %% \ No newline at end of file +%% +%% This is file `setstack.sty', created by VIK 15 Dec 1998 +%% Reproduces useful macros from amsmath.sty, thus avoiding the need to load the entire +%% amsmath.sty and run into conflicts +%% Adds definitions for \overset, \underset, \sideset, \substack, \boxed, \leftroot, +%% \uproot, \dddot, \ddddot, \varrow, \harrow (see The LateX Companion, pp 225-227) +\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) +[1998/12/15]% LaTeX date must December 1998 or later +\ProvidesPackage{setstack} +\DeclareRobustCommand{\text}{% + \ifmmode\expandafter\text@\else\expandafter\mbox\fi} + \let\nfss@text\text + \def\text@#1{\mathchoice + {\textdef@\displaystyle\f@size{#1}}% + {\textdef@\textstyle\tf@size{\firstchoice@false #1}}% + {\textdef@\textstyle\sf@size{\firstchoice@false #1}}% + {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}% + \check@mathfonts +} +\def\textdef@#1#2#3{\hbox{{% + \everymath{#1}% + \let\f@size#2\selectfont #3}}} + +% adds underset, overset, sideset and +% substack features from amsmath.sty (Companion p. 226) + +\def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag} +\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}} +\def\default@tag{\let\tag\dft@tag} + +\def\Let@{\let\\\math@cr} +\def\restore@math@cr{\def\math@cr@@@{\cr}} + +\def\overset#1#2{\binrel@{#2}% +\binrel@@{\mathop{\kern\z@#2}\limits^{#1}}} +\def\underset#1#2{\binrel@{#2}% +\binrel@@{\mathop{\kern\z@#2}\limits_{#1}}} +\def\sideset#1#2#3{% +\@mathmeasure\z@\displaystyle{#3}% +\global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@ + \setbox\tw@\box\@ne +\@mathmeasure4\displaystyle{\copy\tw@#1}% +\@mathmeasure6\displaystyle{#3\nolimits#2}% +\dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@ +\hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}% +} + +\newenvironment{subarray}[1]{% +\vcenter\bgroup +\Let@ \restore@math@cr \default@tag +\baselineskip\fontdimen10 \scriptfont\tw@ +\advance\baselineskip\fontdimen12 \scriptfont\tw@ +\lineskip\thr@@\fontdimen8 \scriptfont\thr@@ +\lineskiplimit\lineskip +\ialign\bgroup\ifx c#1\hfil\fi +$\m@th\scriptstyle##$\hfil\crcr +}{% +\crcr\egroup\egroup +} +\newcommand{\substack}[1]{\subarray{c}#1\endsubarray} + +% definitions of dddot and ddddot(p. 225) + +\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ + \hbox{\normalfont ...}\vss}}}} +\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ + \hbox{\normalfont....}\vss}}}} + +% definitions of leftroot, uproot (p.225) + +\begingroup \catcode`\"=12 +\gdef\@@sqrt#1{\radical"270370 {#1}} +\endgroup + +\def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha} +\def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha} +\newcount\uproot@ +\newcount\leftroot@ +\def\root{\relaxnext@ + \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else + \ifx\@let@token\leftroot\let\next@\nextiii@\else + \let\next@\plainroot@\fi\fi\next@}% + \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}% + \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else + \DN@.{\FN@\nextv@}\fi\next@.}% + \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else + \let\next@\plainroot@\fi\next@}% + \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}% + \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}% + \def\nextvii@{\ifx\@let@token\@sptoken + \DN@. {\FN@\nextviii@}\else + \DN@.{\FN@\nextviii@}\fi\next@.}% + \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else + \let\next@\plainroot@\fi\next@}% + \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}% + \bgroup\uproot@\z@\leftroot@\z@\FN@\next@} +\def\plainroot@#1\of#2{\setbox\rootbox\hbox{% + $\m@th\scriptscriptstyle{#1}$}% + \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}} + {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup} +\def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}% + \dimen@\ht\z@\advance\dimen@-\dp\z@ + \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}% + \advance\dimen@ by1.667\wd\@ne + \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox + \mkern-10mu\mkern\leftroot@ mu\boxz@} + +% definition of \boxed (math in frame, no dollars) p 225 + +\def\boxed#1{\fbox{\m@th$\displaystyle#1$}} + +% definition of \smash for top and bottom parts of expression in braces (p.227) +\renewcommand{\smash}[2][tb]{% + \def\smash@{#1}% + \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else + \@xp\makesm@sh\fi{#2}} + +% additional difinitions for arrows +% zero mm wide #2 mm long vertical arrow shifted 1 mm to left, 1 mm up +\newcommand{\varrow}[2]{% +\unitlength=1mm + \begin{picture}(0,6) % (0,0) + % - for #1 to point arrow down,+ to point arrow up + \end{picture}% + \put(0,6){\vector(0,#1 3){#2}} % 6,- for down, 0, + for up +} +% 1 mm high #2 mm long horizontal arrow shifted 1 mm up (0,-1) +\newcommand{\harrow}[2]{% +\unitlength=1mm + \begin{picture}(8,1)(0,-1) % + % 1 mm distance between arrow and stackreled object over it (8,1) + \put(0,0){\vector(#1 2,0){#2}} % - to point arrow left, + right + \end{picture}% +} + +\endinput +%% end of setstack.sty +%% Corrections history: +%% diff --git a/stokes.tex b/stokes.tex index cfec1f2..06fea32 100644 --- a/stokes.tex +++ b/stokes.tex @@ -40,6 +40,50 @@ \maketitle +\section{Preamble} + +Complex landscapes are basically functions of many variables having many minima and, inevitably, +many saddles of all indices, i.e. the number of unstable directions. Optimization problems +require us to find the deepest minima, often a difficult task. +For example, particles with a repulsive mutual potential enclosed in a box will have many stable +configurations, and we are asked to find the one with lowest energy. + +An aim of complexity theory is to be able to classify these landscapes in families having +common properties. Two simplifications make the task potentially tractable. The first is to consider +the limit of many variables. In the example of the particles, the limit of many particles (i.e. the thermodynamic limit) +may be expected to bring about simplifications. +The second simplification is of more technical nature: we consider functions that contain some random element to them, and we +study the average of an ensemble. The paradigm of this is the spin-glass, where the interactions are random, and we are asked to find the +ground state energy {\em on average over randomness}. + +Spin glass theory gave a surprise: random landscapes come in two kinds: those that +have a `threshold level' of energy, below which there are many minima but no saddles, +separated by high barriers, and those that have all sorts of saddles all the way down to the lowest +energy levels, and local minima are separated by relatively small barriers. +The latter are still complex, but good solutions are easier to find. +This classification is closely related to the structure of their Replica Trick solutions. +Armed with this solvable (random) example, it was easy to find non-random examples +that behave, at least approximately, in these two ways (e.g. sphere packings and the Travelling salesman Problem, +belong to first and second classes, respectively). + +What about systems whose variables are not real, but rather complex? +Recalling the Cauchy-Riemann conditions, we immediately see a difficulty: if our cost is, say, the real part of a function of +$N$ complex variables, in terms of the corresponding $2N$ real variables it has only saddles of index $N$! +Even worse: often not all saddles are equally interesting, so simply finding the lowest is not usually what we +need to do (more about this below). +As it turns out, there is a set of interesting questions to ask, as we describe below. For each saddle, there is a `thimble' +spanned by the +lines along which the cost function decreases. The way in which these thimbles fill the complex space is crucial for many +problems of analytic continuation, and is thus what we need to study. The central role played by saddles +in a real landscape, the `barriers', is now played by the Stokes lines, where thimbles exchange their properties. +Perhaps not surprisingly, the two classes of real landscapes described above retain their role in the complex case, but now +the distinction is that while in the first class the Stokes lines are rare, in the second class they proliferate. + +In this paper we shall start from a many-variable integral of a real function, and deform it in the many variable complex space. +The landscape one faces is the full one in this space, and we shall see that this is an example where the proliferation -- or lack of it -- +of Stokes lines is the interesting quantity in this context. + + \section{Introduction} Analytic continuation of physical theories is sometimes useful. Some theories -- cgit v1.2.3-70-g09d2